OFFSET
0,5
COMMENTS
The Motzkin step set is U=(1,1), H=(1,0) and D=(1,-1). An excursion is a path starting at (0,0), ending on the x-axis and never crossing the x-axis, i.e., staying at nonnegative altitude.
FORMULA
G.f.: ((1+t)(1+t-2t^4-(1+t)*sqrt(1-4t^4)))/(2t^5).
a(4n)=2*C(n), a(4n-1)=C(n), a(4n+1)=C(n) and a(4n+2)=0, where C(n) are the Catalan numbers A000108.
D-finite with recurrence: (n+5)*(3*n^2-27*n+92)*a(n) +16*(3*n-19)*a(n-1) +16*(-3*n+22)*a(n-2) +16*(3*n-25)*a(n-3) -4*(n-3)*(3*n^2-21*n+68)*a(n-4)=0. - R. J. Mathar, Jan 09 2020
EXAMPLE
a(8)=4 since we have the following four excursions of length 8: UHDHUHDH, HUHDHUHD, UHUHDHDH and HUHUHDHD.
CROSSREFS
KEYWORD
nonn,walk
AUTHOR
Valerie Roitner, Nov 29 2019
STATUS
approved